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March  2015, 20(2): 339-371. doi: 10.3934/dcdsb.2015.20.339

On the Boltzmann equation for charged particle beams under the effect of strong magnetic fields

Mihai Bostan 1,

1. 

Institut de Mathématiques de Marseille UMR 7353, Aix Marseille Université, CNRS, Centrale Marseille, 13453 Marseille, France

Received  September 2013 Revised  September 2014 Published  January 2015

The subject matter of this paper concerns the paraxial approximation for the transport of charged particles. We focus on the magnetic confinement properties of charged particle beams. The collisions between particles are taken into account through the Boltzmann kernel. We derive the magnetic high field limit and we emphasize the main properties of the averaged Boltzmann collision kernel, together with its equilibria.
Keywords: Particle beams, Boltzmann equation, finite Larmor radius regime, $H$-theorem..
Mathematics Subject Classification: Primary: 35Q75, 82D10; Secondary: 78A3.
Citation: Mihai Bostan. On the Boltzmann equation for charged particle beams under the effect of strong magnetic fields. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 339-371. doi: 10.3934/dcdsb.2015.20.339
References:
[1]

M. Bostan, The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime, Asymptot. Anal., 61 (2009), 91-123.  Google Scholar

[2]

M. Bostan, Transport equations with disparate advection fields. Application to the gyrokinetic models in plasma physics, J. Differential Equations, 249 (2010), 1620-1663. doi: 10.1016/j.jde.2010.07.010.  Google Scholar

[3]

M. Bostan, Gyrokinetic Vlasov equation in three dimensional setting. Second order approximation, SIAM J. Multiscale Model. Simul., 8 (2010), 1923-1957. doi: 10.1137/090777621.  Google Scholar

[4]

M. Bostan and C. Caldini-Queiros, Finite Larmor radius approximation for collisional magnetized plasmas, C. R. Math. Acad. Sci. Paris, 350 (2012), 879-884. doi: 10.1016/j.crma.2012.09.019.  Google Scholar

[5]

M. Bostan and C. Caldini-Queiros, Finite Larmor radius approximation for collisional magnetic confinement. Part I: The linear Boltzmann equation, Quart. Appl. Math., 72 (2014), 323-345. doi: 10.1090/S0033-569X-2014-01356-1.  Google Scholar

[6]

M. Bostan and C. Caldini-Queiros, Finite Larmor radius approximation for collisional magnetic confinement. Part II: The Fokker-Planck-Landau equation,, to appear in Quart. Appl. Math., ().   Google Scholar

[7]

A. J. Brizard, A guiding-center Fokker-Planck collision operator for nonuniform magnetic fields, Phys. Plasmas, 11 (2004), 4429-4438. doi: 10.1063/1.1780532.  Google Scholar

[8]

A. J. Brizard and T. S. Hahm, Foundations of nonlinear gyrokinetic theory, Rev. Modern Phys., 79 (2007), 421-468. doi: 10.1103/RevModPhys.79.421.  Google Scholar

[9]

C. Cercignani, The Boltzmann Equation and Its Applications, Springer-Verlag New-York 1988. doi: 10.1007/978-1-4612-1039-9.  Google Scholar

[10]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, 106, Springer-Verlag New-York, 1994. doi: 10.1007/978-1-4419-8524-8.  Google Scholar

[11]

R. C. Davidson and H. Qin, Physics of Charged Particle Beams in High Energy Accelerators, Imperial College Press, World Scientific Singapore, 2001. doi: 10.1142/p250.  Google Scholar

[12]

P. Degond and P.-A. Raviart, On the paraxial approximation of the stationary Vlasov-Maxwell system, Math. Models Meth. Appl. Sci., 3 (1993), 513-562. doi: 10.1142/S0218202593000278.  Google Scholar

[13]

F. Filbet and E. Sonnendrücker, Modeling and numerical simulation of space charge dominated beams in the paraxial approximation, Math. Models Methods Appl. Sci., 16 (2006), 763-791. doi: 10.1142/S0218202506001340.  Google Scholar

[14]

E. Frénod, Application of the averaging method to the gyrokinetic plasma, Asymptot. Anal., 46 (2006), 1-28.  Google Scholar

[15]

E. Frénod and A. Mouton, Two-dimensional finite Larmor radius approximation in canonical gyrokinetic coordinates, J. Pures Appl. Math. Adv. Appl., 4 (2010), 135-169.  Google Scholar

[16]

E. Frénod and E. Sonnendrücker, Homogenization of the Vlasov equation and of the Vlasov-Poisson system with strong external magnetic field, Asymptotic Anal., 18 (1998), 193-213.  Google Scholar

[17]

E. Frénod and E. Sonnendrücker, The finite Larmor radius approximation, SIAM J. Math. Anal., 32 (2001), 1227-1247. doi: 10.1137/S0036141099364243.  Google Scholar

[18]

X. Garbet, G. Dif-Pradalier, C. Nguyen, Y. Sarazin, V. Grandgirard and Ph. Ghendrih, Neoclassical equilibrium in gyrokinetic simulations, Phys, Plasmas, 16 (2009), 062503. doi: 10.1063/1.3153328.  Google Scholar

[19]

F. Golse and L. Saint-Raymond, The Vlasov-Poisson system with strong magnetic field, J. Math. Pures Appl., 78 (1999), 791-817. doi: 10.1016/S0021-7824(99)00021-5.  Google Scholar

[20]

G. Laval, S. Mas-Gallic and P.-A. Raviart, Paraxial approximation of ultra-relativistic intense beams, Numer. Math., 69 (1994), 33-60. doi: 10.1007/s002110050079.  Google Scholar

[21]

D. Levermore, Moment closure hierarchies for kinetic theories, J. Statist. Phys., 83 (1996), 1021-1065. doi: 10.1007/BF02179552.  Google Scholar

[22]

J. Madsen, Gyrokinetic linearized Landau collision operator, Phys. Review, 87 (2013), 011101. doi: 10.1103/PhysRevE.87.011101.  Google Scholar

[23]

P.-A. Raviart, Paraxial approximation of the stationary Vlasov-Maxwell equations, in Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar, Vol. XIII, Paris, (1991-1993), Pitman Res. Notes Math. Ser., 302, Longman Sci. Tech., Harlow, 1994, 158-171.  Google Scholar

[24]

H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules,, Z. Wahrsch. Verw. Gebiele, 46 (): 67.  doi: 10.1007/BF00535689.  Google Scholar

[25]

G. Toscani and C. Villani, Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas, J. Statist. Phys., 94 (1999), 619-637. doi: 10.1023/A:1004589506756.  Google Scholar

[26]

G. Toscani and C. Villani, Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation, Comm. Math. Phys., 203 (1999), 667-706. doi: 10.1007/s002200050631.  Google Scholar

[27]

C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Rational Mech. Anal., 143 (1998), 273-307. doi: 10.1007/s002050050106.  Google Scholar

[28]

C. Villani, Contribution à l'étude mathématique des collisions en théorie cinétique, Master's thesis, Université Paris-Dauphine France, 2000. Google Scholar

[29]

X. Q. Xu and M. N. Rosenbluth, Numerical simulation of ion-temperature-gradient-driven modes, Phys. Fluids B, 3 (1991), 627-643. doi: 10.1063/1.859862.  Google Scholar

show all references

References:
[1]

M. Bostan, The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime, Asymptot. Anal., 61 (2009), 91-123.  Google Scholar

[2]

M. Bostan, Transport equations with disparate advection fields. Application to the gyrokinetic models in plasma physics, J. Differential Equations, 249 (2010), 1620-1663. doi: 10.1016/j.jde.2010.07.010.  Google Scholar

[3]

M. Bostan, Gyrokinetic Vlasov equation in three dimensional setting. Second order approximation, SIAM J. Multiscale Model. Simul., 8 (2010), 1923-1957. doi: 10.1137/090777621.  Google Scholar

[4]

M. Bostan and C. Caldini-Queiros, Finite Larmor radius approximation for collisional magnetized plasmas, C. R. Math. Acad. Sci. Paris, 350 (2012), 879-884. doi: 10.1016/j.crma.2012.09.019.  Google Scholar

[5]

M. Bostan and C. Caldini-Queiros, Finite Larmor radius approximation for collisional magnetic confinement. Part I: The linear Boltzmann equation, Quart. Appl. Math., 72 (2014), 323-345. doi: 10.1090/S0033-569X-2014-01356-1.  Google Scholar

[6]

M. Bostan and C. Caldini-Queiros, Finite Larmor radius approximation for collisional magnetic confinement. Part II: The Fokker-Planck-Landau equation,, to appear in Quart. Appl. Math., ().   Google Scholar

[7]

A. J. Brizard, A guiding-center Fokker-Planck collision operator for nonuniform magnetic fields, Phys. Plasmas, 11 (2004), 4429-4438. doi: 10.1063/1.1780532.  Google Scholar

[8]

A. J. Brizard and T. S. Hahm, Foundations of nonlinear gyrokinetic theory, Rev. Modern Phys., 79 (2007), 421-468. doi: 10.1103/RevModPhys.79.421.  Google Scholar

[9]

C. Cercignani, The Boltzmann Equation and Its Applications, Springer-Verlag New-York 1988. doi: 10.1007/978-1-4612-1039-9.  Google Scholar

[10]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, 106, Springer-Verlag New-York, 1994. doi: 10.1007/978-1-4419-8524-8.  Google Scholar

[11]

R. C. Davidson and H. Qin, Physics of Charged Particle Beams in High Energy Accelerators, Imperial College Press, World Scientific Singapore, 2001. doi: 10.1142/p250.  Google Scholar

[12]

P. Degond and P.-A. Raviart, On the paraxial approximation of the stationary Vlasov-Maxwell system, Math. Models Meth. Appl. Sci., 3 (1993), 513-562. doi: 10.1142/S0218202593000278.  Google Scholar

[13]

F. Filbet and E. Sonnendrücker, Modeling and numerical simulation of space charge dominated beams in the paraxial approximation, Math. Models Methods Appl. Sci., 16 (2006), 763-791. doi: 10.1142/S0218202506001340.  Google Scholar

[14]

E. Frénod, Application of the averaging method to the gyrokinetic plasma, Asymptot. Anal., 46 (2006), 1-28.  Google Scholar

[15]

E. Frénod and A. Mouton, Two-dimensional finite Larmor radius approximation in canonical gyrokinetic coordinates, J. Pures Appl. Math. Adv. Appl., 4 (2010), 135-169.  Google Scholar

[16]

E. Frénod and E. Sonnendrücker, Homogenization of the Vlasov equation and of the Vlasov-Poisson system with strong external magnetic field, Asymptotic Anal., 18 (1998), 193-213.  Google Scholar

[17]

E. Frénod and E. Sonnendrücker, The finite Larmor radius approximation, SIAM J. Math. Anal., 32 (2001), 1227-1247. doi: 10.1137/S0036141099364243.  Google Scholar

[18]

X. Garbet, G. Dif-Pradalier, C. Nguyen, Y. Sarazin, V. Grandgirard and Ph. Ghendrih, Neoclassical equilibrium in gyrokinetic simulations, Phys, Plasmas, 16 (2009), 062503. doi: 10.1063/1.3153328.  Google Scholar

[19]

F. Golse and L. Saint-Raymond, The Vlasov-Poisson system with strong magnetic field, J. Math. Pures Appl., 78 (1999), 791-817. doi: 10.1016/S0021-7824(99)00021-5.  Google Scholar

[20]

G. Laval, S. Mas-Gallic and P.-A. Raviart, Paraxial approximation of ultra-relativistic intense beams, Numer. Math., 69 (1994), 33-60. doi: 10.1007/s002110050079.  Google Scholar

[21]

D. Levermore, Moment closure hierarchies for kinetic theories, J. Statist. Phys., 83 (1996), 1021-1065. doi: 10.1007/BF02179552.  Google Scholar

[22]

J. Madsen, Gyrokinetic linearized Landau collision operator, Phys. Review, 87 (2013), 011101. doi: 10.1103/PhysRevE.87.011101.  Google Scholar

[23]

P.-A. Raviart, Paraxial approximation of the stationary Vlasov-Maxwell equations, in Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar, Vol. XIII, Paris, (1991-1993), Pitman Res. Notes Math. Ser., 302, Longman Sci. Tech., Harlow, 1994, 158-171.  Google Scholar

[24]

H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules,, Z. Wahrsch. Verw. Gebiele, 46 (): 67.  doi: 10.1007/BF00535689.  Google Scholar

[25]

G. Toscani and C. Villani, Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas, J. Statist. Phys., 94 (1999), 619-637. doi: 10.1023/A:1004589506756.  Google Scholar

[26]

G. Toscani and C. Villani, Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation, Comm. Math. Phys., 203 (1999), 667-706. doi: 10.1007/s002200050631.  Google Scholar

[27]

C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Rational Mech. Anal., 143 (1998), 273-307. doi: 10.1007/s002050050106.  Google Scholar

[28]

C. Villani, Contribution à l'étude mathématique des collisions en théorie cinétique, Master's thesis, Université Paris-Dauphine France, 2000. Google Scholar

[29]

X. Q. Xu and M. N. Rosenbluth, Numerical simulation of ion-temperature-gradient-driven modes, Phys. Fluids B, 3 (1991), 627-643. doi: 10.1063/1.859862.  Google Scholar

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